188 research outputs found
Joint spectral embeddings of random dot product graphs
Multiplex networks describe a set of entities, with multiple relationships among them, as a collection of networks over a common vertex set. Multiplex networks naturally describe complex systems where units connect across different modalities whereas single network data only permits a single relationship type. Joint spectral embedding methods facilitate analysis of multiplex network data by simultaneously mapping vertices in each network to points in Euclidean space, entitled node embeddings, where statistical inference is then performed. This mapping is performed by spectrally decomposing a matrix that summarizes the multiplex network. Different methods decompose different matrices and hence yield different node embeddings. This dissertation analyzes a class of joint spectral embedding methods which provides a foundation to compare these different approaches to multiple network inference.
We compare joint spectral embedding methods in three ways. First, we extend the Random Dot Product Graph model to multiplex network data and establish the statistical properties of node embeddings produced by each method under this model. This analysis facilitates a full bias-variance analysis of each method and uncovers connections between these methods and methods for dimensionality reduction. Second, we compare the accuracy of algorithms which utilize these different node embeddings in a variety of multiple network inference tasks including community detection, vertex anomaly detection, and graph hypothesis testing. Finally, we perform a time and space complexity analysis of each method and present a case study in which we analyze interactions between New England sports fans on the social news aggregation and discussion website, Reddit. These findings provide a theoretical and practical guide to compare joint spectral embedding techniques and highlight the benefits and drawbacks of utilizing each method in practice
Limit results for distributed estimation of invariant subspaces in multiple networks inference and PCA
We study the problem of estimating the left and right singular subspaces for
a collection of heterogeneous random graphs with a shared common structure. We
analyze an algorithm that first estimates the orthogonal projection matrices
corresponding to these subspaces for each individual graph, then computes the
average of the projection matrices, and finally finds the matrices whose
columns are the eigenvectors corresponding to the largest eigenvalues of
the sample averages. We show that the algorithm yields an estimate of the left
and right singular vectors whose row-wise fluctuations are normally distributed
around the rows of the true singular vectors. We then consider a two-sample
hypothesis test for the null hypothesis that two graphs have the same edge
probabilities matrices against the alternative hypothesis that their edge
probabilities matrices are different. Using the limiting distributions for the
singular subspaces, we present a test statistic whose limiting distribution
converges to a central (resp. non-central ) under the null
(resp. alternative) hypothesis. Finally, we adapt the theoretical analysis for
multiple networks to the setting of distributed PCA; in particular, we derive
normal approximations for the rows of the estimated eigenvectors using
distributed PCA when the data exhibit a spiked covariance matrix structure
Group Invariance, Stability to Deformations, and Complexity of Deep Convolutional Representations
The success of deep convolutional architectures is often attributed in part
to their ability to learn multiscale and invariant representations of natural
signals. However, a precise study of these properties and how they affect
learning guarantees is still missing. In this paper, we consider deep
convolutional representations of signals; we study their invariance to
translations and to more general groups of transformations, their stability to
the action of diffeomorphisms, and their ability to preserve signal
information. This analysis is carried by introducing a multilayer kernel based
on convolutional kernel networks and by studying the geometry induced by the
kernel mapping. We then characterize the corresponding reproducing kernel
Hilbert space (RKHS), showing that it contains a large class of convolutional
neural networks with homogeneous activation functions. This analysis allows us
to separate data representation from learning, and to provide a canonical
measure of model complexity, the RKHS norm, which controls both stability and
generalization of any learned model. In addition to models in the constructed
RKHS, our stability analysis also applies to convolutional networks with
generic activations such as rectified linear units, and we discuss its
relationship with recent generalization bounds based on spectral norms
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